Two classes of machine learning approaches that have seen success in modeling nonlinear systems include reservoir networks [1]}, [2]} and recurrent neural networks (RNN) [3]}, [4]}, [5]}. Both of these methods work by finding the optimal parameters to some function that maps \(u(t)\)  forward to some \(\tilde{u}(t+\tau )\)  by using a history of passed states. The parameters are then trained by minimizing the difference between the prediction and the true state \(\left<||u(t+\tau )-\tilde{u}(t+\tau )||^2\right>\) . Both methods can predict multiple Lyapunov times for the chaotic dynamics of the KSE
[5]}. The Lyapunov time is the relevant time scale for chaotic systems and will be discussed in more detail below. Some drawbacks of these methods are they typically result in an increase in the dimension of the problem, they typically require evenly spaced data, and the start up typically needs multiple known states because these methods are non-Markovian (e.g. predictions of the next state require multiple past states).
